330 research outputs found
Adaptive isogeometric analysis with hierarchical box splines
Isogeometric analysis is a recently developed framework based on finite
element analysis, where the simple building blocks in geometry and solution
space are replaced by more complex and geometrically-oriented compounds. Box
splines are an established tool to model complex geometry, and form an
intermediate approach between classical tensor-product B-splines and splines
over triangulations. Local refinement can be achieved by considering
hierarchically nested sequences of box spline spaces. Since box splines do not
offer special elements to impose boundary conditions for the numerical solution
of partial differential equations (PDEs), we discuss a weak treatment of such
boundary conditions. Along the domain boundary, an appropriate domain strip is
introduced to enforce the boundary conditions in a weak sense. The thickness of
the strip is adaptively defined in order to avoid unnecessary computations.
Numerical examples show the optimal convergence rate of box splines and their
hierarchical variants for the solution of PDEs
The Argyris isogeometric space on unstructured multi-patch planar domains
Multi-patch spline parametrizations are used in geometric design and
isogeometric analysis to represent complex domains. We deal with a particular
class of planar multi-patch spline parametrizations called
analysis-suitable (AS-) multi-patch parametrizations (Collin,
Sangalli, Takacs; CAGD, 2016). This class of parametrizations has to satisfy
specific geometric continuity constraints, and is of importance since it allows
to construct, on the multi-patch domain, isogeometric spaces with optimal
approximation properties. It was demonstrated in (Kapl, Sangalli, Takacs; CAD,
2018) that AS- multi-patch parametrizations are suitable for modeling
complex planar multi-patch domains.
In this work, we construct a basis, and an associated dual basis, for a
specific isogeometric spline space over a given AS-
multi-patch parametrization. We call the space the Argyris
isogeometric space, since it is across interfaces and at all
vertices and generalizes the idea of Argyris finite elements to tensor-product
splines. The considered space is a subspace of the entire
isogeometric space , which maintains the reproduction
properties of traces and normal derivatives along the interfaces. Moreover, it
reproduces all derivatives up to second order at the vertices. In contrast to
, the dimension of does not depend on the domain
parametrization, and admits a basis and dual basis which possess
a simple explicit representation and local support.
We conclude the paper with some numerical experiments, which exhibit the
optimal approximation order of the Argyris isogeometric space and
demonstrate the applicability of our approach for isogeometric analysis
A family of quadrilateral finite elements
We present a novel family of quadrilateral finite elements, which
define global spaces over a general quadrilateral mesh with vertices of
arbitrary valency. The elements extend the construction by (Brenner and Sung,
J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product
degree , to all degrees . Thus, we call the family of
finite elements Brenner-Sung quadrilaterals. The proposed quadrilateral
can be seen as a special case of the Argyris isogeometric element of (Kapl,
Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar
degrees of freedom as the classical Argyris triangles. Just as for the Argyris
triangle, we additionally impose continuity at the vertices. In this
paper we focus on the lower degree cases, that may be desirable for their lower
computational cost and better conditioning of the basis: We consider indeed the
polynomial quadrilateral of (bi-)degree~, and the polynomial degrees
and by employing a splitting into or polynomial
pieces, respectively.
The proposed elements reproduce polynomials of total degree . We show that
the space provides optimal approximation order. Due to the interpolation
properties, the error bounds are local on each element. In addition, we
describe the construction of a simple, local basis and give for
explicit formulas for the B\'{e}zier or B-spline coefficients of the basis
functions. Numerical experiments by solving the biharmonic equation demonstrate
the potential of the proposed quadrilateral finite element for the
numerical analysis of fourth order problems, also indicating that (for )
the proposed element performs comparable or in general even better than the
Argyris triangle with respect to the number of degrees of freedom
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